Nyquist, aliasing and oversampling: where a tone really lands

The sampling theorem is usually quoted as "you need twice the highest frequency", which is true and nearly useless on its own, because it says nothing about what happens when you break it. Nothing is filtered out by sampling. Frequencies above the limit come back as lower frequencies, perfectly formed and impossible to distinguish from a real signal at that frequency.

The calculator works out where an arbitrary input frequency lands, how many samples per cycle it gets, what oversampling is worth, and which timing noise limit applies at your frequency.

The theorem, and its two misstatements

A signal whose bandwidth is B can be reconstructed exactly from samples taken faster than 2B per second. The two conditions are load-bearing: the signal really must contain nothing above B, and the rate must be strictly greater than the limit. A 20 kHz band sampled at exactly 40 kHz fails the second condition, and a band containing a 21 kHz component fails the first no matter how high the rate.

The commonly repeated version — "you need twice the highest frequency" — reverses the requirement. The rate does not adapt to the signal; the signal must be band-limited to half the rate you have chosen.

Folding, precisely

Sampling a tone at f produces aliases at |f − k · fs| for every integer k. The useful consequence is that the alias pattern repeats every fs, so reducing f modulo the sample rate is enough to see what will happen: a 25 kHz tone in a 48 kHz system appears at 48 − 25 = 23 kHz, and a 49 kHz tone appears at 1 kHz.

Frequencies that land at or below Nyquist are the awkward part of the mental model: 25 kHz and 23 kHz are not merely similar readings, they are the same sequence of samples. No processing afterwards can separate them, which is why the anti-alias filter has to sit before the converter and has to be analog.

Oversampling, and what it is worth

Quantisation noise is spread evenly from DC to Nyquist, and its total power depends only on the bit depth. Widen the band and the same total noise is spread thinner: the in-band noise falls in proportion to the oversampling ratio, so the SNR gain is 10 · log10(ratio), where the ratio is sample rate divided by twice the bandwidth of interest.

Each doubling of the ratio is worth 3.01 dB, and a ratio of four is worth 6.02 dB — one extra bit. That is the entire reason a 1-bit converter running at 2.8 MHz can deliver 20 bits of audio: the noise has been pushed outside the band, and the digital filter that follows throws the band away. It is also why "oversampling" in a converter datasheet is a noise-shaping claim rather than a resolution claim.

Measured against the ideal figures, a 16-bit converter is limited to 98.09 dB and a 24-bit one to 146.25 dB. Very few 24-bit converters reach their theoretical floor in practice, and jitter is usually why.

The jitter limit, which depends on frequency

A sampling clock error of σ_t RMS turns into a voltage error proportional to the signal's rate of change, so the resulting noise floor is approximately

Note that amplitude does not appear. Jitter noise scales with frequency and with the signal's slope, not with its level, which makes it a fixed floor that a quiet passage cannot hide. With 1 ns of RMS jitter the floor is 104 dB at 1 kHz but only 78 dB at 20 kHz — a 26 dB penalty for moving up two decades, which is exactly what the 20 dB per decade slope predicts.

The practical number to carry around: at 20 kHz, a 24-bit converter's 146 dB theoretical floor needs jitter below about 5 ps. No ordinary clock achieves that, so high-resolution conversion at high frequency is a clock problem before it is a converter problem.

Rates you will actually meet

• 44.1 kHz — Nyquist at 22.05 kHz, a deliberately awkward number inherited from video line rates; 2.267 µs per sample.
• 48 kHz — Nyquist 24 kHz, 20.83 µs per sample, 48 samples per cycle at 1 kHz. The default for everything that will not inherit a video clock.
• 96 kHz and 192 kHz — chosen for easier anti-alias filtering and for margin against jitter in the analogue path, not for bandwidth nobody can hear. Note that the jitter limit tightens by 6 dB and 12 dB respectively if the *signal* frequency stays the same; what improves is the filter, not the clock sensitivity.
• Undersampling — deliberately sampling a band-pass signal at less than twice its centre frequency is legal and common in radio, provided the whole band fits inside one Nyquist zone. The folding arithmetic above is the same; only the intent changes.

Five mistakes worth avoiding

1. Treating Nyquist as a brick wall. It is a fold line — content above it comes back, mirrored.
2. Filtering after the converter. Once folded, an alias is indistinguishable from a real tone; the anti-alias filter must be analog and must come first.
3. Quoting the sample rate as the bandwidth. A 48 kHz system carries 24 kHz, and in practice less, because the anti-alias filter needs a transition band.
4. Expecting oversampling to buy bits without noise shaping. Oversampling alone buys 3.01 dB per doubling; the large numbers come from shaping the noise out of the band.
5. Forgetting that jitter is frequency-dependent. A clock that is fine for a 1 kHz measurement can be the dominant error at 20 kHz.

Summary

Reduce the input frequency modulo the sample rate to find the alias, remember that the alias is bit-identical to a real signal at that frequency, and put the anti-alias filter before the converter. Then check the clock: at high frequencies, jitter usually sets the floor long before the converter does.

Values shown are engineering aids rather than measurements; see the disclaimer and the tool index.